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Underwater communication systems are critical to scientific and civil missions in the ocean. High speed underwater image transmission capabilities can enable the next generation of undersea expeditions. However, current acoustic communication technologies cannot support image transmission since the ocean is a challenging communication channel. We propose a new discrete-time all-analog-processing system that combines compressive sensing (CS) techniques with nonlinear mapping as analog joint source-channel codes for the ocean channel. The CS processing generates minimum amount of information necessary for transmission. Further, it can efficiently compress the images through the exploitation of their sparsity and statistical distribution in the wavelet domain. Nonlinear mapping via space-filling curves provides protection for the CS measurements against channel distortions. Therefore, underwater images are compressed and directly coded in the analog domain, completely skipping the digitization process. It employs the conventional minimum mean squared error (MMSE) equalizer to compensate for the intersymbol interference (ISI). Simulations over the measured acoustic channel in the ocean have demonstrated reliable image transmission results.